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SL 5.3—Differentiating polynomials, n E Z - IB Questionbank | RevisionDojo

Practice SL 5.3—Differentiating polynomials, n E Z with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider the function $g(x) = e^x \cos(x)$ .

Find the derivative of the function $g(x) = e^x \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Find the equation of the normal line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Question 2

SL & HLPaper 1

Consider the function $f(x) = \sin(x) + \cos(x)$ , where $x$ is in radians.

Find the derivative of the function $f(x) = \sin(x) + \cos(x)$ .

[2]

Determine the critical points of the function $f(x) = \sin(x) + \cos(x)$ in the interval $[0, 2\pi]$ .

[3]

Classify the critical points found in Part 2 as local maxima, minima, or inflection points.

[4]

Question 3

SLPaper 1

An engineer is designing a section of a roller coaster track, modeled by the function $f(x)=bx^3-8x^2$ , where $x$ represents the horizontal distance along the track (in meters) from the start, and $f(x)$ gives the vertical height of the track relative to the station platform.

The slope of the track affects the speed and forces on the coaster. At a specific point $P(2,y)$ along the track, the slope is measured to be $4$ . This information is used to determine the value of the constant $b$ .

Find the expression for the slope of the track at any point $x$ by calculating $f'(x)$ .

[2]

Given that the slope of the track at point $P(2,y)$ is $4$ , find the value of the constant $b$ .

[2]

Question 4

SLPaper 1

Consider the function $f(x) = 3x^4 - 8x^3 + 6x^2 - 4x + 1$ .

Find the first derivative $f'(x)$ .

[2]

Find the second derivative $f''(x)$ .

[2]

Find the x-coordinates of the points of inflection of the function.

[3]

Question 5

SLPaper 1

Let $g(x) = \sin(x) + \cos(x)$ .

Find the derivative of the function $g(x)$ .

[1]

Find the equation of the tangent line to the graph of $g(x)$ at $x = \frac{\pi}{4}$ .

[3]

Hence, or otherwise, show that at $x=\frac{\pi}{4}$ the function attains its maximum.

[4]

Question 6

SL & HLPaper 1

Given the function $g(x) = \tan(x)$ , where $x$ is in radians and $x \neq \frac{\pi}{2} + n\pi$ , $n \in \mathbb{Z}$ .

Find the derivative of the function $g(x) = \tan(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = \tan(x)$ at the point where $x = \frac{\pi}{4}$ .

[4]

Question 7

HLPaper 1

This question involves the introduction of differential calculus and focuses on finding the derivative of a function using first principles and applying basic differentiation rules.

Consider the function $f(x) = 3x^2 - 2x + 4$ . Using the definition of the derivative, find $f'(x)$ .

[5]

Using the result from part 1, find the slope of the tangent to the curve $y = f(x)$ at the point where $x = 1$ .

[2]

Find the equation of the tangent line to the curve $y = f(x)$ at the point where $x = 1$ .

[3]

Determine the x-coordinate where the tangent to the curve $y = f(x)$ is horizontal.

[2]

Question 8

HLPaper 1

Consider the function $f(x) = x^2 \sin(x) + \cos(x)$ .

Find the derivative of the function $f(x) = x^2 \sin(x) + \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $f(x) = x^2 \sin(x) + \cos(x)$ at the point where $x = \frac{\pi}{2}$ .

[2]

Find the equation of the normal line to the curve $f(x) = x^2 \sin(x) + \cos(x)$ at the point where $x = \frac{\pi}{2}$ .

[2]

Question 9

SLPaper 1

An architect is designing a roller coaster with a section of track modeled by the function $f(x)=2x^3-8x^2+10x+5$ , where $x$ represents the horizontal distance along the track (in meters), and $f(x)$ gives the height of the track at any point.

To ensure safety and thrill, the architect needs to know the slope of the track at specific points. The slope at $x=2$ will help the architect understand the steepness at this part of the ride.

Find the expression for the slope of the roller coaster track at any point $x$ .

[3]

Calculate the slope of the ride at $x=2$ .

[1]

Question 10

SLPaper 1

This question involves understanding the basics of differential calculus and applying the concept of derivatives to solve problems.

Consider the function $f(x) = x^3 - 3x^2 + 2x + 5$ .

Find the derivative of the function $f(x)$ with respect to $x$ .

[2]

Determine the critical points of the function $f(x)$ .

[3]

Determine whether each critical point is a local maximum, local minimum, or neither.

[3]

Sketch the graph of the function $f(x) = x^3 - 3x^2 + 2x + 5$ , indicating the critical points and their nature.

[3]

SL 5.3—Differentiating polynomials, n E Z Questionbank